Compact Subsets of T3 Spaces
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Theorem
Let $T = \left({X, \vartheta}\right)$ be a $T_3$ space.
Let $A \subseteq X$ be compact in $T$.
Then for each $U \in \vartheta$ such that $A \subseteq U$:
- $\exists V \in \vartheta: A \subseteq V \subseteq V^- \subseteq U$
where $V^-$ denotes the closure of $V$.
Proof
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Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 3$: Compactness Properties and the $T_i$ Axioms
